For a taste of the Mayan Ruins, try the ruins site in
the Hotel District. The price is very reasonable, as
in cheap and public transportation via the bus is
available both ways. The site is known as El Rey.
A day trip on your own to Isla Mujeres utilizing the
public ferries can be very interesting and as long as
you stay in the North end of the Isla, most things are
in walking distance with reasonable prices for beaches,
shopping, drinks and eateries.
Another trip that is a bit long by public bus would be the
ruins in Tulum, the park entrance fee is very cheap and
swimming is available on site. You could do this trip and
make a stop in Playa Del Carmen, by public bus again,
for snacks, shopping, drinks and exploring before
returning back to Cancun.
Visiting downtown Cancun can be adventurous for the
novice and easily had by bus. Interesting shopping,
cheaper local restaurants and cantinas are the rule.